TRANSPORT PROCESSES AND THERMAL CONDUCTIVITY NEAR THE CRITICAL TEMPERATURE

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TRANSPORT PROCESSES AND THERMAL CONDUCTIVITY NEAR THE CRITICAL

TEMPERATURE

G. Dixon, J. Rives, D. Walton

To cite this version:

G. Dixon, J. Rives, D. Walton. TRANSPORT PROCESSES AND THERMAL CONDUCTIVITY NEAR THE CRITICAL TEMPERATURE. Journal de Physique Colloques, 1971, 32 (C1), pp.C1- 528-C1-530. �10.1051/jphyscol:19711177�. �jpa-00213999�

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JOURNAL DE PHYSIQUE Colloque C 1, supplément au n° 2-3, Tome 32, Février-Mars 1971, page C l - 528

TRANSPORT PROCESSES AND THERMAL CONDUCTIVITY NEAR THE CRITICAL TEMPERATURE (*)

G. S. DIXON (**)

Oak Ridge National Laboratory, Oak Ridge, Tenn.

J. E. RIVES

University of Georgia, Athens, Ga.

and D. WALTON

McMaster University, Hamilton, Ont. Canada

Résumé. — Près d'une transition magnétique, une forte interaction entre les fluctuations critiques du système de spins et les phonons peut conduire à une diffusion de ceux-ci, avec un effet important sur la conductibilité thermique du réseau. La diffusion par le système de spins peut être séparée de la diffusion non magnétique, par l'application d'un champ magnétique assez intense pour éliminer les fluctuations en alignant les spins.

Par cette technique, des résultats ont été obtenus sur les ferromagnétiques d'Heinsenberg CUK2CI4 : 2 H2O, CuRbzCU : 2H2O, CIJCNH-OÎ Br4 : 2H2O et l'antiferromagnétique MnCla :4H20. Nous avons évalué des paramètres de transport des spins tels que le temps de relaxation critique et la conductivité thermique des spins. Cette dernière est trouvée inférieure de plusieurs ordres de grandeur à la conductivité thermique par les phonons. Grâce à la haute fréquence des phonons thermiques (~ 10¹¹ Hz à 1 °K) nos cristaux sont d'assez bonne qualité pour une observation facile de la transition entre régime hydrodynamique et régime critique.

Abstract. — Near magnetic phase transitions a strong interaction between critical fluctuations in the magnetic spin system and the phonon heat carriers can lead to a phonon scattering that has a marked effect on the lattice thermal conductivity. By applying a large enough magnetic field to impose a ferromagnetic order on the spins, thereby eliminating the critical fluctuations, the phonon scattering by the spin system can be separated from that of non-magnetic origin.

Using this technique results have been obtained for the Heisenberg ferromagnets CUK2CI4 : 2 H20 , CuRbaCU : 2H2O, Cu(NH4>2Br4 : 2H2O, and the antiferromagnet MnCl2 : 4H2O. Such spin transport parameters as the critical relaxation rate for order parameter fluctuations and the spin thermal conductivity have been estimated. The spin thermal conductivity is found to be many orders of magnitude less than the phonon conductivity. Owing to the high frequency of thermal phonons (~ 10J1 Hz at 1 °K) our crystals are of sufficient quality for the effect of the transition from the hydrodynamic to the critical regime to be readily observed.

We have been able to account for the thermal conductivity near a magnetic transition entirely in terms of scattering of phonons by critical fluctuation.

Data have been obtained on jthe insulating [ferromagnets CuK²Cl⁴ : 2 H²0 , CuRb²Cl⁴ : 2 H²0 , Cu(NH4)2Br4 : 2j H20 , and the [antiferromagnet MnCl2 : 4 H20 [2]. The ferromagnets are particularly attractive in that they are almost ideal Heisenberg systems.

These results have been obtained by employing a technique which involves the measurement of thermal conductivities both in zero field, and in a magnetic field large enough to superimpose a ferromagnetic

order on the spin system. This separates the effect of the critical fluctuations from the scattering of non- magnetic origin. By observing the field dependence of the conductivity it is also possible to measure the contribution of the spin system to the heat flow, and the non-critical scattering of phonons by the spin system [1].

To the best of our knowledge these results are the only measurements available for insulating ferromagnets. Furthermore, the coupling constants for these systems are too weak for the effects to be observable in a conventional ultrasonic attenuation experiment.

(*) Research sponsored by the U. S. Atomic Energy Com- mission under contract with the Union Carbide Corporation.

(**) U. S. A. E. C. Postdoctoral Fellow under appointment from Oak Ridge Associated Universities.

The experimental technique has been described in detail elsewhere [1, 2] as have the results for

M n C l²4 H²0 and CuK²Cl⁴ : 2 H²0 . Figures 1 and 2 display our results for the other two ferromagnets.

What do we learn from these measurements ? Well at the outset we find [1] that the spin system does not conduct a measurable amount of heat. This may be surprising to some in view of the large specific heat of the spin system relative to the phonon system for these crystals. However near Ta the rate at which energy can diffuse through the spin system is so low that this mode of transport is rendered ineffective. In fact this is what is meant by critical slowing down.

Our results confirm theoretical estimates which suggest that this contribution is negligible [3, 4].

Since we can neglect transport by the spins we can confine our attention to the phonons. The factor which determines the conductivity will be the rate at which phonons are scattered. We can evaluate the effect of uninteresting sources of scattering due to the boundaries and defects from the high field results.

These can be accounted for by a boundary resistance which is close to the Casimir length plus a small defect term.

We now turn to the magnetic scattering which, within the accuracy of our measurements, is all due to critical fluctuations. We will only consider T > Te,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711177

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TRANSPORT PROCESSES AND THERMAL CONDUCTIVITY NEAR THE CRITICAL TEMPERATURE C 1 - ⁵²⁹

FIG. 1. - Thermal conductivity as a function of temperature for CuRbzClc : 2 H20.

FIG. 2. - Thermal conductivity as a function of temperature for Cu(NH4)2Br4 : 2 HzO.

the region below T, cannot be fitted in a very satis- factory manner. In the hydrodynamic region the theories of Laramore and Kadanoff [4] and Kawasaki 151 yield for the relaxation time

where q is the phonon wave-vector, E = (1 - TIT,) and p for a Heisenberg ferromagnet is close to 513.

For T

<

T, we can fit our data to this expression, and for CuK2C14 : 2 H 2 0 p = 1.6

+

^0.10,^for

CuRb2C14 2 H 2 0 p = 1.7 _+ 0.2 and for

These values are obviously in excellent agreement with theory. However, just as obviously this cannot be the whole story, for if equation (1) were correct the conductivity would be zero at T,. The value of E at which the expression begins to break down can readily be estimated from the point where the calculated conductivity begins to fall appreciably below the measured value. In other words, the scattering law must change form at this value of ^8,^8,to preserve agreement with experiment.

What is the reason for this behaviour ? For all three salts ^E,

-

^lo-', which is too large for it to be a rounding of the critical transition due to the presence of impurities. The most obvious other possibility is that a transition from the hydrodynamic region to the critical region is occurring, as suggested by Dixon and Walton [I]. This could be due to the critical frequency for spin fluctuations becoming lower than a typical phonon frequency (-- 10" Hz). It is unlikely to be due to the spin-spin correlation length becoming of the order of a typical phonon wavelength (-- 1 000

A),

as suggested by Dixon and Walton, as the correlation length would appear to be at least an order of magnitude larger than would be expected.

Following Laramore and Kadanoff [4] we can estimate a lower limit for the critical frequency very roughly as follows :

For theseysalts T,

-

1 OK so that [s:],=~ w lo1* H z ,

using S: cc E + we conclude that this condition is ~ ~ ~

violated. In fact if relation (2) is even approximately correct, then this condition must always be violated above T, in a thermal conductivity experiment since the important phonon frequencies lie between kT/2 and 6 kT.

Kawasaki [5] has derived an expression for the ultrasonic attenuation which is valid for frequencies higher than the critical frequency provided q t 4 1, where

5

is the correlation length for critical fluctuations of the order parameter. The following expression approaches Kawasaki's relation asymptotically as

Using this expression we can obtain good agreement between theory and experiment.

The spin-phonon coupling constant we deduce from our measurements is also close to the value obtained from thermal expansion measurements [6] for

I t is interesting that the coupling constant for the antiferromagnet should be of the same order of magnitude, but the scattering is much less. It appears

36

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C 1

-

⁵³⁰ ^G.S. DIXON, J. E. RIVES AND D. WALTON

that this is probably due to the coupling being strictly we have observed phonon scattering in the critical nearest neighbour, which results in zero attenuation, region. Although the analysis of the results of thermal as Luthi and Pollina point out [7]. conductivity data is rendered uncertain by the necessity In conclusion we wish to observe that we have of integrating over a range of phonon frequencies, been able to obtain data using thermal conductivity, our results provide some experimental justification for which is unavailable by other means. In particular Kawasaki's expression.

References

[I] DIXON (G. S.) and WALTON (D.), Phys. Rev., 1969, [4] LARAMORE (G. E.) and KADANOFF (L. P.), Phys. Rev.

185, 735. 1969, 187, 619.

[2] RIVES (J. E.), WALTON@.), and Drxo~(G. S.), J. A. P., ¹⁵³K A ~ A ~ A ~ (K.1, J. A- P., 1970, 41, 1311- 1970, 41, 1435. [61 F'HILP (J. W.) and ADAMS (E. D.), Preprint.

[3] HUBER (D. L.), Solid State Comm., 1968, 6 , 685. ^[7]LUTHI @.) and POLLINA (R. J.), Phys. Rev. Letters, 1969, 22, 717.